Nash Equilibrium

game-theory

Definition

Nash Equilibrium

A Nash equilibrium is a strategy profile in a game where no player can improve their payoff by changing only their own strategy.

For players $N=\{1,\dots ,n\}$, strategy sets $S_i$, payoff functions $u_i:S_1\times \cdots \times S_n\to \mathbb{R}$, and strategy profile $s^{\ast }=(s_1^{\ast },\dots ,s_n^{\ast })$, $s^{\ast }$ is a Nash equilibrium iff

$$\forall i\in N,\forall s_i\in S_i:u_i(s_i^{\ast },s_{-i}^{\ast })\ge u_i(s_i,s_{-i}^{\ast }).$$

Here $s_{-i}^{\ast }$ denotes the equilibrium strategies of all players except player $i$.